In-Depth Notes on Interest Rate and Bond Valuation

Bond Definitions

  • Bond: A debt contract.
  • Interest-only loan: A type of loan where only interest payments are made before paying back the principal.
  • Par value (Face value) - Usually $1,000.
  • Coupon rate (c): Interest rate specified by the issuer as a percentage of par value (F).
  • Coupon payment (C): Amount received periodically calculated as C=cimesFC = c imes F.
  • Time to maturity (t): Duration from valuation date to bond maturity.
  • Yield-to-maturity (YTM) (r): The market rate of return on similar bonds; this is determined by the bond's price.

Valuation - Present Value of Future Cash Flows

  • General formula: Bond valuation is based on discounting future cash flows (C, face value) at the appropriate discount rate (YTM).
  • Bond pricing determines: The present value (PV) of expected future cash flows from holding the bond at a market-determined rate (opportunity cost).

Bond Valuation Scenarios

  • At a Discount: When YTM > Coupon Rate (c) → Price (P) < Face Value (F).
    • Example: 10% coupon rate, YTM = 11%, Price = $963.04
  • At a Premium: When YTM < Coupon Rate (c) → Price (P) > Face Value (F).
    • Example: 10% coupon rate, YTM = 8%, Price = $1,196.36
  • At Par: When YTM = Coupon Rate (c) → Price (P) = Face Value (F).
    • Example: 10% coupon rate, YTM = 10%, Price = $1,000.

Bond Pricing Equation with Cash Flows

  • Price computation: B=PV(extAnnuity)+PV(extLumpSum)B = PV( ext{Annuity}) + PV( ext{Lump Sum})
    • Where:
    • Coupons (C) or (C=cimesFC = c imes F)
    • Face Value (F)

Yield-to-Maturity (YTM) Calculations

  • General YTM reference:
    • If bond priced below par: YTM > c.
    • If bond priced above par: YTM < c.
  • Calculation methods:
    • Financial calculator input
    • Excel function: =PV(rate, nper, pmt, fv, type)

Interest Rate Risks

  • Price Risk: Price volatility caused by changing interest rates.
    • Long-term bonds exhibit higher price risk than short-term bonds.
  • Reinvestment Rate Risk: Risks related to the reinvestment of coupon payments at uncertain future rates.
    • Short-term bonds have higher reinvestment risk.

Bond Ratings and Investment Quality

  • Investment Grade Bonds:
    • High Grade: Aaa (Moody's), AAA (S&P) - Extremely strong payment capacity.
    • Medium Grade: Baa (Moody's), BBB (S&P) - Adequate capacity, more vulnerable to changes.
  • Speculative Bonds:
    • Low Grade: B rated bonds.
    • Very Low Grade: C rated bonds indicate default risk.

Types of Bonds

  • Government Bonds: Include T-bills (short-term), T-notes (medium-term), and T-bonds (long-term).
  • Zero Coupon Bonds: No periodic payments, entire yield is capital gain.
    • Example: Treasury Bills, U.S. Savings Bonds.

Tax Considerations

  • Comparing Returns:
    • Taxable bond yield after tax: 8 ext{%} imes (1 - 0.4) = 4.8 ext{%} .
    • Municipal bond yield: 6%. Preferred if municipal yield is higher.
    • Break-even tax rate for equivalency: T = 25 ext{%} .

Inflation Impact on Interest Rates

  • Real rate of interest: Measures change in purchasing power.
  • Nominal rate of interest: Includes inflation effects.
  • The Fisher Effect: Relationship expressed as (1+R)=(1+r)(1+h)(1 + R) = (1 + r)(1 + h), where R is nominal rate, r is real rate, and h is inflation rate.

Term Structure of Interest Rates

  • Represents the relationship between time to maturity and yield.
  • Yield Curve: Upward sloping (normal) and downward sloping (inverted).